Fast Subsystem Research Articles

Velocity-Free Trajectory Tracking and Active Vibration Control of Flexible Space Manipulator

This article studies the trajectory tracking and active vibration control of flexible space manipulator without velocity measurements. Variable-speed control moment gyros (VSCMGs) are adopted as actuators for vibration suppression. Based on the singular perturbation method, the dynamics model of the manipulator system considering the uncertainties and disturbances is decoupled into two reduced-order subsystems: the slow subsystem and the fast subsystem. For the slow subsystem, neural network (NN) is used to approximate the unknown nonlinear functions. An NN-based adaptive integral sliding-mode observer is designed to estimate some unknown velocity variables. An observer-based slow controller is proposed. Considering the uncertainties and the unmeasured modal information of the fast system, a modified extended state observer (ESO) is designed for the estimation of the uncertainties and unknown fast variables. On the basis of the modified ESO, an adaptive feedback controller is proposed for the flexible motion. Lyapunov theory is used to prove the stability of the proposed controllers. The weighted robust pseudoinverse steering laws are designed for VSCMGs. The effectiveness of the proposed observer-based hybrid controller is verified via numerical simulations.

Finite-region stability of 2-D singular Roesser systems with directional delays

In this paper, the problem of finite-region stability is studied for a class of two-dimensional (2-D) singular systems described by using the Roesser model with directional delays. Based on the regularity, we first decompose the underlying singular 2-D systems into fast and slow subsystems corresponding to dynamic and algebraic parts. Then, with the Lyapunov-like 2-D functional method, we construct a weighted 2-D functional candidate and utilize zero-type free matrix equations to derive delay-dependent stability conditions in terms of linear matrix inequalities (LMIs). More specifically, the derived conditions ensure that all state trajectories of the system do not exceed a prescribed threshold over a pre-specified finite region of time for any initial state sequences when energy-norms of dynamic parts do not exceed given bounds.

Rearrangement of Energy Levels between Energy Super-Bands Characterized by Second Chern Class

We generalize the dynamical analog of the Berry geometric phase setup to the quaternionic model of Avron et al. In our dynamical quaternionic system, the fast half-integer spin subsystem interacts with a slow two-degrees-of-freedom subsystem. The model is invariant under the 1:1:2 weighted SO(2) symmetry and spin inversion. There is one formal control parameter in addition to four dynamical variables of the slow subsystem. We demonstrate that the most elementary qualitative phenomenon associated with the rearrangement of the energy super-bands of our model consists of the rearrangement of one energy level between two energy superbands which takes place when the formal control parameter takes the special isolated value associated with the conical degeneracy of the semi-quantum eigenvalues. This qualitative phenomenon is of topological origin, and is characterized by the second Chern class of the associated semi-quantum system. The correspondence between the number of redistributed energy levels and the second Chern number is confirmed through a series of examples.

A unified physiological framework of transitions between seizures, sustained ictal activity and depolarization block at the single neuron level

The majority of seizures recorded in humans and experimental animal models can be described by a generic phenomenological mathematical model, the Epileptor. In this model, seizure-like events (SLEs) are driven by a slow variable and occur via saddle node (SN) and homoclinic bifurcations at seizure onset and offset, respectively. Here we investigated SLEs at the single cell level using a biophysically relevant neuron model including a slow/fast system of four equations. The two equations for the slow subsystem describe ion concentration variations and the two equations of the fast subsystem delineate the electrophysiological activities of the neuron. Using extracellular K+ as a slow variable, we report that SLEs with SN/homoclinic bifurcations can readily occur at the single cell level when extracellular K+ reaches a critical value. In patients and experimental models, seizures can also evolve into sustained ictal activity (SIA) and depolarization block (DB), activities which are also parts of the dynamic repertoire of the Epileptor. Increasing extracellular concentration of K+ in the model to values found during experimental status epilepticus and DB, we show that SIA and DB can also occur at the single cell level. Thus, seizures, SIA, and DB, which have been first identified as network events, can exist in a unified framework of a biophysical model at the single neuron level and exhibit similar dynamics as observed in the Epileptor.Author Summary: Epilepsy is a neurological disorder characterized by the occurrence of seizures. Seizures have been characterized in patients in experimental models at both macroscopic and microscopic scales using electrophysiological recordings. Experimental works allowed the establishment of a detailed taxonomy of seizures, which can be described by mathematical models. We can distinguish two main types of models. Phenomenological (generic) models have few parameters and variables and permit detailed dynamical studies often capturing a majority of activities observed in experimental conditions. But they also have abstract parameters, making biological interpretation difficult. Biophysical models, on the other hand, use a large number of variables and parameters due to the complexity of the biological systems they represent. Because of the multiplicity of solutions, it is difficult to extract general dynamical rules. In the present work, we integrate both approaches and reduce a detailed biophysical model to sufficiently low-dimensional equations, and thus maintaining the advantages of a generic model. We propose, at the single cell level, a unified framework of different pathological activities that are seizures, depolarization block, and sustained ictal activity.

Trajectory control and vibration suppression of rigid‐flexible parallel robot based on singular perturbation method

AbstractTo accurately, stably, and efficiently control the complex rigid‐flexible coupled robot, this paper takes Delta robot as the study object and decomposes it into slow subsystem and fast subsystem in different timescales by using singular perturbation principle, which the slow subsystem representing the rigid motion and the fast subsystem representing flexible vibration. In addition, the backstepping control system is designed for the slow subsystem, and the dynamic surface control system based on K‐observer is designed for the fast subsystem, and the overall control scheme is proposed by combining the backstepping control system and the dynamic surface control system, and the stability of the proposed overall control scheme is proven. Finally, the proposed control scheme is compared with proportional derivative control and trajectory control with workspace lattices, and the related experimental results are analyzed in details.

Firing patterns and bifurcation analysis of Purkinje cell dendrite model

This paper mainly studied firing patterns and related bifurcations in the Purkinje cell dendrite model. Based on the methods of equivalent potentials and time scale analysis, the initial six-dimensional (6D) dendrite model is reduced to a 3D form to facilitate the calculation. We numerically show that the dendrite model could exhibit period-adding bifurcation and four bursting patterns for several vital parameters. Then the bifurcation mechanisms and transition of these four bursting patterns are discussed by phase plane analysis, and two-parameter bifurcation analysis of the fast subsystem, respectively. Moreover, we computed the first Lyapunov coefficient to determine the stability of Hopf bifurcation. Ultimately, we analyzed the codimension-two bifurcation of the whole system and gave a detailed theoretical derivation of the Bogdanov–Takens bifurcation.

Observer-based event-triggered control of linear system with two-time scales

In this paper, the stabilization of singularly perturbed systems by means of observer-based event-triggered control is studied. To obtain the ultimately bounded stability of the closed-loop system, two event-triggering conditions are designed, respectively, for the slow and the fast subsystems of the singularly perturbed systems based on the state errors between the observer and models. In each event-triggering condition, there are two different stages to be designed. In the first stage, the event-triggering conditions are given based on the observer error dynamics. While in the second stage, the absolute type event-triggering conditions are used. Under the designed event-triggering conditions, Zeno behavior can be eliminated from the closed-loop systems. Finally, numerical examples are provided to illustrate the efficiency and feasibility of the theoretical results.

Amplitude-modulated spiking as a novel route to bursting: Coupling-induced mixed-mode oscillations by symmetry breaking.

Mixed-mode oscillations consisting of alternating small- and large-amplitude oscillations are increasingly well understood and are often caused by folded singularities, canard orbits, or singular Hopf bifurcations. We show that coupling between identical nonlinear oscillators can cause mixed-mode oscillations because of symmetry breaking. This behavior is illustrated for diffusively coupled FitzHugh-Nagumo oscillators with negative coupling constant, and we show that it is caused by a singular Hopf bifurcation related to a folded saddle-node (FSN) singularity. Inspired by earlier work on models of pancreatic beta-cells [Sherman, Bull. Math. Biol. 56, 811 (1994)], we then identify a new type of bursting dynamics due to diffusive coupling of cells firing action potentials when isolated. In the presence of coupling, small-amplitude oscillations in the action potential height precede transitions to square-wave bursting. Confirming the hypothesis from the earlier work that this behavior is related to a pitchfork-of-limit-cycles bifurcation in the fast subsystem, we find that it is caused by symmetry breaking. Moreover, we show that it is organized by a FSN in the averaged system, which causes a singular Hopf bifurcation. Such behavior is related to the recently studied dynamics caused by the so-called torus canards.

About switched affine system interconnected with fast LTI dynamics

We study an interconnection between a slow switched affine system and a fast LTI system. The lower dimensional slow and fast subsystems are computed via the singular perturbation approach. We consider the classical stabilization method based on the existence of a stable convex combination (while ignoring the fast dynamics) and we provide an LMI condition for checking ultimate boundedness when the fast dynamics is taken into account. Numerical example illustrates the main result.

Voltage Control of Flexible-Joint Robot Manipulators using Singular Perturbation Technique for Model Order Reduction

Background and Objectives: Robot manipulator with flexible-joints is very complex nonlinear system whose control is one of the most challenging issues in the control word. Therefore, the design of voltage-based controller in addition, using of order reduction methods can reduce the complexity of the control law in such systems. Methods: This paper proposes a voltage controller for flexible-joint robot manipulators based on singular perturbation method. The presented control approach has all three advantages of the singular perturbation method for model order reduction, the proper structure of Port-Hamiltonian systems, and voltage control strategy (VCS). In this approach, the robot manipulator model is divided into three sub-systems, slow, medium and fast sub-systems. Each of the sub-systems is controlled using separate controller. In addition, the stability of these sub-systems and ultimately the whole system are proved. Unlike other related works, in this work the tracking error system is considered from the beginning, and by singular perturbation method, a controller is designed to stabilize the tracking system. Moreover, in the suggested voltage-based controller unlike torque control strategy, the electrical model of actuators is used. Results: The main advantages of proposed approach are simple structure, using only velocity of motors, the position of the joints as a control signal and considering the electrical model of the actuators. So, practical implementation of this controller will be with much less effort, compared to the methods like feedback linearization or other controllers in related works. Moreover, using the Lyapunov-based method, the ultimate bounded stability of the closed loop system is proved. Then, some simulations are provided for tracking, regulation, robustness and response speed purposes. Conclusion: Since, the controllers for every sub-system are designed separately, also, the control signal parameters such as joints position, motors velocity, and motors current can be simply measured, therefore, the structure of designed controller is very simple and practically implementable. As the simulation results confirm, the performance of this controller is appropriate, even when external disturbances are present or the frequency of reference signal increases. At last, by comparison and analysis of the simulation results between the presented approach and a related work, the suitable tracking performance of the suggested controller is shown.

Analysis to dynamics of complex electrical activities in Wilson model of brain neocortical neuron using fast-slow variable dissection with two slow variables

The neocortex of the brain plays a most important role in achieving functions of the brain via the electrical activities of neurons. Understanding the transition regularity of firing patterns and underlying dynamics of firing patterns of neurons can help to identify the brain functions and to treat some brain diseases. Different neocortical neurons exhibit regular spiking (RS), fast spiking (FS), intrinsic bursting (IB), and continuous bursting (CB), which play vital roles and wide range of functions. Fast-slow variable dissection method combined with bifurcation analysis has been an effective method to identify the underlying dynamical mechanism of spiking and bursting modulated by a single slow variable. The spiking is related to the stable limit cycle of the fast subsystem, and the bursting is associated with the transitions or bifurcations between the stable limit cycle and resting state of the fast subsystem. Such underlying dynamics of bursting has been widely used to distinguish different bursting patterns and identify complex dynamics of bursting modulated by various different factors such as synaptic current, autaptic current, and stimulations applied at a suitable phase related to the bifurcations, which play important roles in the real nervous system to regulate neural firing behaviors. Unfortunately, the bursting of neocortical neuronal model (wilson model) is modulated by two slow variables, i.e. the gating variable of calcium-activated potassium channel <i>H</i> and the gating variable of T-type calcium channel <i>T,</i> with <i>H</i> being slower than <i>T</i>. Then, the underlying dynamical mechanism of the IB and CB of the neocortical neurons cannot be acquired by the fast-slow variable dissection method when <i>H</i> is taken as the sole slow variable, due to the fact that the fast-subsystem contains the slow variable <i>T</i>. In the present paper, we use the fast-slow variable dissection method with two slow variables (<i>H</i> and <i>T </i>) to analyze the bursting patterns. The bifurcations of the fast subsystem, and the intersections between the bifurcation curves and the phase trajectory of bursting in the parameter plane (<i>H</i>, <i>T </i>) are acquired. Owing to the fact that neither of the two slow variables of the bursting is sufficiently slow, the bifurcations of only some intersections are related to the bursting behaviors, but others not. Then, the position relationship between the bifurcation curves and bursting trajectory in the three-dimensional space (<i>H</i>, <i>T</i>, <i>V </i>) (<i>V</i> is membrane potential of bursting) is further acquired, from which the bifurcations related to bursting behaviors are acquired and bifurcations unrelated to bursting behaviors are excluded. The start phase and the termination phase of the burst of the IB are related to the saddle-node on invariant circle (SNIC) bifurcation, but not to the saddle-node (SN) bifurcation. The start phase and termination phase of the burst of the CB are related to the SNIC bifurcation and the supercritical Andronov-Hopf (SupHopf) bifurcation, respectively, but not to the SN bifurcation. The results present a comprehensive and in-depth understanding of the underlying dynamics of bursting patterns in the neocortical neurons, thereby laying the foundation for regulating the firing patterns of the neocortical neurons.

Integrated Fixed Time Sliding Mode Control for Motion and Vibration of Space Robot with Fully Flexible Base–Link–Joint

The dynamic modeling, motion control and flexible vibration active suppression of space robot under the influence of flexible base, flexible link and flexible joint are explored, and motion and vibration integrated fixed-time sliding mode control of fully flexible system is designed. The flexibility of the base and joints are equivalent to the vibration effect of linear springs and torsion springs. The flexible links are regarded as Euler–Bernoulli simply supported beams, which are analyzed by the hypothetical mode method, and the dynamic model of the fully flexible space robot is established by using the Lagrange equation. Then, the singular perturbation theory is used to decompose the model into slow subsystem including rigid motion and the link flexible vibrations, and fast subsystems including the base and the joint flexible vibrations. A fixed time sliding mode control based on hybrid trajectory is designed for the slow subsystem to ensure that the base and joints track the desired trajectory in a limited time while achieving vibration suppression on the flexible links. For the fast subsystem, linear quadratic optimal control is used to suppress the flexible vibration of the base and joints. The simulation results show that the controller proposed in the paper can make the system state converge within a fixed time, is robust to model uncertainty and external interference, and can effectively suppress the flexible vibration of the base, links, and joints.

Chaos in the Rulkov Neuron Model Based on Marotto’s Theorem

The existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem. Firstly, the stability conditions of the model are briefly renewed through analyzing the eigenvalues of the model, which are very important preconditions for the existence of a snap-back repeller. Secondly, the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast–slow dynamics technique, in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods, such as waveforms, Lyapunov exponents, and bifurcation diagrams. Thirdly, for the two-dimensional Rulkov neuron model, it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces, which pave a new way to identify the existence of a snap-back repeller.

Fast-Slow Coupling Dynamics Behavior of the van der Pol-Rayleigh System

In this paper, the dynamic behavior of the van der Pol-Rayleigh system is studied by using the fast–slow analysis method and the transformation phase portrait method. Firstly, the stability and bifurcation behavior of the equilibrium point of the system are analyzed. We find that the system has no fold bifurcation, but has Hopf bifurcation. By calculating the first Lyapunov coefficient, the bifurcation direction and stability of the Hopf bifurcation are obtained. Moreover, the bifurcation diagram of the system with respect to the external excitation is drawn. Then, the fast subsystem is simulated numerically and analyzed with or without external excitation. Finally, the vibration behavior and its generation mechanism of the system in different modes are analyzed. The vibration mode of the system is affected by both the fast and slow varying processes. The mechanisms of different modes of vibration of the system are revealed by the transformation phase portrait method, because the system trajectory will encounter different types of attractors in the fast subsystem.

Fast–slow variable dissection with two slow variables related to calcium concentrations: a case study to bursting in a neural pacemaker model

Neuronal bursting is an electrophysiological behavior participating in physiological or pathological functions and a complex nonlinear behavior alternating between burst and quiescent state modulated by slow variables. Identification of dynamics of bursting modulated by two slow variables is still an open problem. In the present paper, a novel fast–slow variable dissection method with two slow variables is proposed to analyze the complex bursting simulated in a four-dimensional neuronal model to describe firing associated with pathological pain. The lumenal (\(C_\mathrm{lum}\)) and intracellular (\(C_\mathrm{in}\)) calcium concentrations are the slowest variables, respectively, in the quiescent state and burst duration. Questions encountered when the traditional method with one slow variable is used. When \(C_\mathrm{lum}\) is taken as slow variable, the burst is successfully identified to terminate near the saddle-homoclinic bifurcation point of the fast subsystem and begin not from the saddle-node bifurcation. With \(C_\mathrm{in}\) chosen as slow variable, \(C_\mathrm{lum}\) value of the initiation point of burst is far from the saddle-node bifurcation point, due to \(C_\mathrm{lum}\) not contained in the equation of the membrane potential. To overcome this problem, both \(C_\mathrm{in}\) and \(C_\mathrm{lum}\) are regarded as slow variables; the two-dimensional fast subsystem exhibits a saddle-node bifurcation point, which is extended to a saddle-node bifurcation curve by introducing the \(C_\mathrm{lum}\) dimension. Then, the initial point of burst is successfully identified to be near the saddle-node bifurcation curve. The results present a feasible method for fast–slow variable dissection and a deep understanding of the complex bursting behavior with two slow variables, which is helpful for the modulation to pathological pain.

Two-Time-Scale Approach to Characterize the Steady-State Security Region for the Electricity-Gas Integrated Energy System

The knowledge of the steady-state security region is essential for secure operation of the integrated energy system (IES). The two-time-scale feature of the IES is explored to facilitate construction of its exact steady-state security region. Conditions allowing decomposition of the IES into fast and slow subsystems are established. A complete characterization of the IES steady-state security region via the stable equilibrium manifold of the fast subsystem and the slow subsystem is derived and theoretically proven. This characterization can significantly facilitate construction of the IES steady-state security region (SSR) and is derived under the AC power flow model and gas flow model without any model approximation and simplification. Numerical studies are conducted to validate the derived characterization of the SSR and to illustrate the advantages of the proposed characterization in terms of robustness and reduced computational burden. Numerical studies show that exploring the fast and slow subsystems can lead to fast computation of feasible solutions with a 20-fold speed up in calculation time, as compared with the direct integration of the original IES system. Hence, the proposed two-time-scale method appears to be fast and yet accurate.

Chattering-Suppressed Sliding Mode Control for Flexible-Joint Robot Manipulators

In this paper, sliding mode tracking control and its chattering suppression method are investigated for flexible-joint robot manipulators with only state measurements of joint actuators. First, within the framework of singular perturbation theory, the control objective of the system is decoupled into two typical tracking aims of a slow subsystem and a fast subsystem. Then, considering lumped uncertainties (including dynamics uncertainties and external disturbances), a composite chattering-suppressed sliding mode controller is proposed, where a smooth-saturation-function-contained reaching law with adjustable saturation factor is designed to alleviate the inherent chattering phenomenon, and a radial basis function neural network (RBFNN)-based soft computing strategy is applied to avoid the high switching gain that leads to chattering amplification. Simultaneously, an efficient extended Kalman filter (EKF) with respect to a new state variable is presented to enable the closed-loop tracking control with neither position nor velocity measurements of links. In addition, an overall analysis on the asymptotic stability of the whole control system is given. Finally, numerical examples verify the superiority of the dynamic performance of the proposed control approach, which is well qualified to suppress the chattering and can effectively eliminate the undesirable effects of the lumped uncertainties with a smaller switching gain reduced by 80% in comparison to that in the controller without RBFNN. The computational efficiency of the proposed EKF increased by about 26%.

Two-time-scale robust output feedback control for aircraft longitudinal dynamics via sliding mode control and high-gain observer

This paper presents output feedback control for longitudinal dynamics of fixed-wing aircraft using singular perturbation techniques. The longitudinal dynamic model is separated into two timescales. A composite feedback control law based on sliding mode control (SMC) is designed for slow and fast subsystems to track the aerodynamic velocity and flight path angle references. Additionally, two separate high-gain observers (HGO) are designed for the slow and fast subsystems to estimate system states. The proposed technique not only simplifies the design but also is computationally efficient. Simulations are performed in MATLAB/Simulink environment with results demonstrating that the proposed output feedback control method robustly achieves trajectory tracking.

Bursting mechanism in a memristive Lorenz based system and function projective synchronization in its-fractional-order form: Digital implementation under ATmega328P microcontroller

In this paper, bifurcation mechanism of bursting oscillations and function projective synchronization of the fractional order form for a memristive Lorenz based system are investigated. One of the key results is the dependance of relaxation time of bursting oscillation on the fractional order derivative. The newly proposed system has a line equilibrium and accordingly, features hidden attractors. When the system is divided into two subsystems labeled slow subsystem(SS) and fast subsystem(FS), stability analysis indicates that bursting oscillations result from the switching of the equilibrium point of the(FS) between stable and unstable states. We refer this class of bursting as ”source/bursting”. The second aspect of this paper deals with the study of fractional order of the system precisely function projective synchronization in a relay topology. By using some standard nonlinear diagnostic tools such as bifurcation diagram, largest Lyapunov exponent, phase space trajectory and power spectrum, the study of the fractional model reveals that the dynamics strongly depends on system parameters , fractional order q as well as on initial conditions. With a set of chosen parameters , the proposed scheme achieves function projective synchronization in a finite time despite the unpredictability of the scaling function. The Results of digital implementation based on a ATmega328P microcontroller are in good agreement with those found by numerical simulations.

New Bursting Patterns in the Lü System Driven by Orthogonal Parametric Excitation and External Excitation

Compared with the general nonlinear system, multi-timescale system has complex dynamical characteristics of bursting oscillations, which has received extensive attention. This paper aims to show and study the bursting oscillations and bifurcation phenomena in the Lü systems driven by orthogonal parametric excitation and external excitation. By using the fast–slow analysis, we were able to get two different fast subsystems corresponding to different bifurcation diagrams when the slow variable increased and decreased. Several novel bursting modes have been discovered, namely, delayed supHopf/cascaded with PD and IPD/Fold/cascaded with PD and IPD/supHopf-Fold bursting, delayed supHopf/supHopf/Fold-Fold bursting and delayed supHopf/cascaded with PD and IPD/supHopf-Fold bursting. Finally, a simulation circuit of the multi-timescale system is established in Multisim, and a real physical circuit is also built. The experimental results effectively verify the research results.